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Theorem axpowndlem2 9622
Description: Lemma for the Axiom of Power Sets with no distinct variable conditions. Revised to remove a redundant antecedent from the consequence. (Contributed by NM, 4-Jan-2002.) (Proof shortened by Mario Carneiro, 6-Dec-2016.) (Revised and shortened by Wolf Lammen, 9-Jun-2019.)
Assertion
Ref Expression
axpowndlem2 (¬ ∀𝑥 𝑥 = 𝑦 → (¬ ∀𝑥 𝑥 = 𝑧 → ∃𝑥𝑦(∀𝑥(∃𝑧 𝑥𝑦 → ∀𝑦 𝑥𝑧) → 𝑦𝑥)))
Distinct variable group:   𝑦,𝑧

Proof of Theorem axpowndlem2
Dummy variable 𝑤 is distinct from all other variables.
StepHypRef Expression
1 zfpow 4975 . . . 4 𝑤𝑦(∀𝑤(𝑤𝑦𝑤𝑧) → 𝑦𝑤)
2 19.8a 2206 . . . . . . . 8 (𝑤𝑦 → ∃𝑧 𝑤𝑦)
3 sp 2207 . . . . . . . 8 (∀𝑦 𝑤𝑧𝑤𝑧)
42, 3imim12i 62 . . . . . . 7 ((∃𝑧 𝑤𝑦 → ∀𝑦 𝑤𝑧) → (𝑤𝑦𝑤𝑧))
54alimi 1887 . . . . . 6 (∀𝑤(∃𝑧 𝑤𝑦 → ∀𝑦 𝑤𝑧) → ∀𝑤(𝑤𝑦𝑤𝑧))
65imim1i 63 . . . . 5 ((∀𝑤(𝑤𝑦𝑤𝑧) → 𝑦𝑤) → (∀𝑤(∃𝑧 𝑤𝑦 → ∀𝑦 𝑤𝑧) → 𝑦𝑤))
76alimi 1887 . . . 4 (∀𝑦(∀𝑤(𝑤𝑦𝑤𝑧) → 𝑦𝑤) → ∀𝑦(∀𝑤(∃𝑧 𝑤𝑦 → ∀𝑦 𝑤𝑧) → 𝑦𝑤))
81, 7eximii 1912 . . 3 𝑤𝑦(∀𝑤(∃𝑧 𝑤𝑦 → ∀𝑦 𝑤𝑧) → 𝑦𝑤)
9 nfnae 2470 . . . . 5 𝑥 ¬ ∀𝑥 𝑥 = 𝑦
10 nfnae 2470 . . . . 5 𝑥 ¬ ∀𝑥 𝑥 = 𝑧
119, 10nfan 1980 . . . 4 𝑥(¬ ∀𝑥 𝑥 = 𝑦 ∧ ¬ ∀𝑥 𝑥 = 𝑧)
12 nfnae 2470 . . . . . 6 𝑦 ¬ ∀𝑥 𝑥 = 𝑦
13 nfnae 2470 . . . . . 6 𝑦 ¬ ∀𝑥 𝑥 = 𝑧
1412, 13nfan 1980 . . . . 5 𝑦(¬ ∀𝑥 𝑥 = 𝑦 ∧ ¬ ∀𝑥 𝑥 = 𝑧)
15 nfv 1995 . . . . . . 7 𝑤(¬ ∀𝑥 𝑥 = 𝑦 ∧ ¬ ∀𝑥 𝑥 = 𝑧)
16 nfnae 2470 . . . . . . . . . 10 𝑧 ¬ ∀𝑥 𝑥 = 𝑦
17 nfcvd 2914 . . . . . . . . . . 11 (¬ ∀𝑥 𝑥 = 𝑦𝑥𝑤)
18 nfcvf 2937 . . . . . . . . . . 11 (¬ ∀𝑥 𝑥 = 𝑦𝑥𝑦)
1917, 18nfeld 2922 . . . . . . . . . 10 (¬ ∀𝑥 𝑥 = 𝑦 → Ⅎ𝑥 𝑤𝑦)
2016, 19nfexd 2329 . . . . . . . . 9 (¬ ∀𝑥 𝑥 = 𝑦 → Ⅎ𝑥𝑧 𝑤𝑦)
2120adantr 466 . . . . . . . 8 ((¬ ∀𝑥 𝑥 = 𝑦 ∧ ¬ ∀𝑥 𝑥 = 𝑧) → Ⅎ𝑥𝑧 𝑤𝑦)
22 nfcvd 2914 . . . . . . . . . . 11 (¬ ∀𝑥 𝑥 = 𝑧𝑥𝑤)
23 nfcvf 2937 . . . . . . . . . . 11 (¬ ∀𝑥 𝑥 = 𝑧𝑥𝑧)
2422, 23nfeld 2922 . . . . . . . . . 10 (¬ ∀𝑥 𝑥 = 𝑧 → Ⅎ𝑥 𝑤𝑧)
2513, 24nfald 2327 . . . . . . . . 9 (¬ ∀𝑥 𝑥 = 𝑧 → Ⅎ𝑥𝑦 𝑤𝑧)
2625adantl 467 . . . . . . . 8 ((¬ ∀𝑥 𝑥 = 𝑦 ∧ ¬ ∀𝑥 𝑥 = 𝑧) → Ⅎ𝑥𝑦 𝑤𝑧)
2721, 26nfimd 1973 . . . . . . 7 ((¬ ∀𝑥 𝑥 = 𝑦 ∧ ¬ ∀𝑥 𝑥 = 𝑧) → Ⅎ𝑥(∃𝑧 𝑤𝑦 → ∀𝑦 𝑤𝑧))
2815, 27nfald 2327 . . . . . 6 ((¬ ∀𝑥 𝑥 = 𝑦 ∧ ¬ ∀𝑥 𝑥 = 𝑧) → Ⅎ𝑥𝑤(∃𝑧 𝑤𝑦 → ∀𝑦 𝑤𝑧))
2918, 17nfeld 2922 . . . . . . 7 (¬ ∀𝑥 𝑥 = 𝑦 → Ⅎ𝑥 𝑦𝑤)
3029adantr 466 . . . . . 6 ((¬ ∀𝑥 𝑥 = 𝑦 ∧ ¬ ∀𝑥 𝑥 = 𝑧) → Ⅎ𝑥 𝑦𝑤)
3128, 30nfimd 1973 . . . . 5 ((¬ ∀𝑥 𝑥 = 𝑦 ∧ ¬ ∀𝑥 𝑥 = 𝑧) → Ⅎ𝑥(∀𝑤(∃𝑧 𝑤𝑦 → ∀𝑦 𝑤𝑧) → 𝑦𝑤))
3214, 31nfald 2327 . . . 4 ((¬ ∀𝑥 𝑥 = 𝑦 ∧ ¬ ∀𝑥 𝑥 = 𝑧) → Ⅎ𝑥𝑦(∀𝑤(∃𝑧 𝑤𝑦 → ∀𝑦 𝑤𝑧) → 𝑦𝑤))
33 nfeqf2 2452 . . . . . . . . 9 (¬ ∀𝑦 𝑦 = 𝑥 → Ⅎ𝑦 𝑤 = 𝑥)
3433naecoms 2465 . . . . . . . 8 (¬ ∀𝑥 𝑥 = 𝑦 → Ⅎ𝑦 𝑤 = 𝑥)
3534adantr 466 . . . . . . 7 ((¬ ∀𝑥 𝑥 = 𝑦 ∧ ¬ ∀𝑥 𝑥 = 𝑧) → Ⅎ𝑦 𝑤 = 𝑥)
3614, 35nfan1 2222 . . . . . 6 𝑦((¬ ∀𝑥 𝑥 = 𝑦 ∧ ¬ ∀𝑥 𝑥 = 𝑧) ∧ 𝑤 = 𝑥)
37 nfnae 2470 . . . . . . . . . . . . . 14 𝑧 ¬ ∀𝑥 𝑥 = 𝑧
38 nfeqf2 2452 . . . . . . . . . . . . . . 15 (¬ ∀𝑧 𝑧 = 𝑥 → Ⅎ𝑧 𝑤 = 𝑥)
3938naecoms 2465 . . . . . . . . . . . . . 14 (¬ ∀𝑥 𝑥 = 𝑧 → Ⅎ𝑧 𝑤 = 𝑥)
4037, 39nfan1 2222 . . . . . . . . . . . . 13 𝑧(¬ ∀𝑥 𝑥 = 𝑧𝑤 = 𝑥)
41 elequ1 2152 . . . . . . . . . . . . . 14 (𝑤 = 𝑥 → (𝑤𝑦𝑥𝑦))
4241adantl 467 . . . . . . . . . . . . 13 ((¬ ∀𝑥 𝑥 = 𝑧𝑤 = 𝑥) → (𝑤𝑦𝑥𝑦))
4340, 42exbid 2247 . . . . . . . . . . . 12 ((¬ ∀𝑥 𝑥 = 𝑧𝑤 = 𝑥) → (∃𝑧 𝑤𝑦 ↔ ∃𝑧 𝑥𝑦))
4443adantll 693 . . . . . . . . . . 11 (((¬ ∀𝑥 𝑥 = 𝑦 ∧ ¬ ∀𝑥 𝑥 = 𝑧) ∧ 𝑤 = 𝑥) → (∃𝑧 𝑤𝑦 ↔ ∃𝑧 𝑥𝑦))
4512, 34nfan1 2222 . . . . . . . . . . . . 13 𝑦(¬ ∀𝑥 𝑥 = 𝑦𝑤 = 𝑥)
46 elequ1 2152 . . . . . . . . . . . . . 14 (𝑤 = 𝑥 → (𝑤𝑧𝑥𝑧))
4746adantl 467 . . . . . . . . . . . . 13 ((¬ ∀𝑥 𝑥 = 𝑦𝑤 = 𝑥) → (𝑤𝑧𝑥𝑧))
4845, 47albid 2246 . . . . . . . . . . . 12 ((¬ ∀𝑥 𝑥 = 𝑦𝑤 = 𝑥) → (∀𝑦 𝑤𝑧 ↔ ∀𝑦 𝑥𝑧))
4948adantlr 694 . . . . . . . . . . 11 (((¬ ∀𝑥 𝑥 = 𝑦 ∧ ¬ ∀𝑥 𝑥 = 𝑧) ∧ 𝑤 = 𝑥) → (∀𝑦 𝑤𝑧 ↔ ∀𝑦 𝑥𝑧))
5044, 49imbi12d 333 . . . . . . . . . 10 (((¬ ∀𝑥 𝑥 = 𝑦 ∧ ¬ ∀𝑥 𝑥 = 𝑧) ∧ 𝑤 = 𝑥) → ((∃𝑧 𝑤𝑦 → ∀𝑦 𝑤𝑧) ↔ (∃𝑧 𝑥𝑦 → ∀𝑦 𝑥𝑧)))
5150ex 397 . . . . . . . . 9 ((¬ ∀𝑥 𝑥 = 𝑦 ∧ ¬ ∀𝑥 𝑥 = 𝑧) → (𝑤 = 𝑥 → ((∃𝑧 𝑤𝑦 → ∀𝑦 𝑤𝑧) ↔ (∃𝑧 𝑥𝑦 → ∀𝑦 𝑥𝑧))))
5211, 27, 51cbvald 2436 . . . . . . . 8 ((¬ ∀𝑥 𝑥 = 𝑦 ∧ ¬ ∀𝑥 𝑥 = 𝑧) → (∀𝑤(∃𝑧 𝑤𝑦 → ∀𝑦 𝑤𝑧) ↔ ∀𝑥(∃𝑧 𝑥𝑦 → ∀𝑦 𝑥𝑧)))
5352adantr 466 . . . . . . 7 (((¬ ∀𝑥 𝑥 = 𝑦 ∧ ¬ ∀𝑥 𝑥 = 𝑧) ∧ 𝑤 = 𝑥) → (∀𝑤(∃𝑧 𝑤𝑦 → ∀𝑦 𝑤𝑧) ↔ ∀𝑥(∃𝑧 𝑥𝑦 → ∀𝑦 𝑥𝑧)))
54 elequ2 2159 . . . . . . . 8 (𝑤 = 𝑥 → (𝑦𝑤𝑦𝑥))
5554adantl 467 . . . . . . 7 (((¬ ∀𝑥 𝑥 = 𝑦 ∧ ¬ ∀𝑥 𝑥 = 𝑧) ∧ 𝑤 = 𝑥) → (𝑦𝑤𝑦𝑥))
5653, 55imbi12d 333 . . . . . 6 (((¬ ∀𝑥 𝑥 = 𝑦 ∧ ¬ ∀𝑥 𝑥 = 𝑧) ∧ 𝑤 = 𝑥) → ((∀𝑤(∃𝑧 𝑤𝑦 → ∀𝑦 𝑤𝑧) → 𝑦𝑤) ↔ (∀𝑥(∃𝑧 𝑥𝑦 → ∀𝑦 𝑥𝑧) → 𝑦𝑥)))
5736, 56albid 2246 . . . . 5 (((¬ ∀𝑥 𝑥 = 𝑦 ∧ ¬ ∀𝑥 𝑥 = 𝑧) ∧ 𝑤 = 𝑥) → (∀𝑦(∀𝑤(∃𝑧 𝑤𝑦 → ∀𝑦 𝑤𝑧) → 𝑦𝑤) ↔ ∀𝑦(∀𝑥(∃𝑧 𝑥𝑦 → ∀𝑦 𝑥𝑧) → 𝑦𝑥)))
5857ex 397 . . . 4 ((¬ ∀𝑥 𝑥 = 𝑦 ∧ ¬ ∀𝑥 𝑥 = 𝑧) → (𝑤 = 𝑥 → (∀𝑦(∀𝑤(∃𝑧 𝑤𝑦 → ∀𝑦 𝑤𝑧) → 𝑦𝑤) ↔ ∀𝑦(∀𝑥(∃𝑧 𝑥𝑦 → ∀𝑦 𝑥𝑧) → 𝑦𝑥))))
5911, 32, 58cbvexd 2437 . . 3 ((¬ ∀𝑥 𝑥 = 𝑦 ∧ ¬ ∀𝑥 𝑥 = 𝑧) → (∃𝑤𝑦(∀𝑤(∃𝑧 𝑤𝑦 → ∀𝑦 𝑤𝑧) → 𝑦𝑤) ↔ ∃𝑥𝑦(∀𝑥(∃𝑧 𝑥𝑦 → ∀𝑦 𝑥𝑧) → 𝑦𝑥)))
608, 59mpbii 223 . 2 ((¬ ∀𝑥 𝑥 = 𝑦 ∧ ¬ ∀𝑥 𝑥 = 𝑧) → ∃𝑥𝑦(∀𝑥(∃𝑧 𝑥𝑦 → ∀𝑦 𝑥𝑧) → 𝑦𝑥))
6160ex 397 1 (¬ ∀𝑥 𝑥 = 𝑦 → (¬ ∀𝑥 𝑥 = 𝑧 → ∃𝑥𝑦(∀𝑥(∃𝑧 𝑥𝑦 → ∀𝑦 𝑥𝑧) → 𝑦𝑥)))
Colors of variables: wff setvar class
Syntax hints:  ¬ wn 3  wi 4  wb 196  wa 382  wal 1629  wex 1852  wnf 1856
This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1870  ax-4 1885  ax-5 1991  ax-6 2057  ax-7 2093  ax-8 2147  ax-9 2154  ax-10 2174  ax-11 2190  ax-12 2203  ax-13 2408  ax-ext 2751  ax-pow 4974
This theorem depends on definitions:  df-bi 197  df-an 383  df-or 835  df-tru 1634  df-ex 1853  df-nf 1858  df-cleq 2764  df-clel 2767  df-nfc 2902
This theorem is referenced by:  axpowndlem3  9623
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